For \(M \in {{M}\mathrm {fd}}\) , let \({\mathrm {Cov}}(M)\)  denote the following category:
an object of \({\mathrm {Cov}}(M)\)  is an open covering \({{U}}= \lbrace U_a\rbrace _{a \in A}\)  of \(M\) , and given another open covering \({{V}}\)  of \(M\)  there is a unique morphism \({{V}}\rightarrow {{U}}\)  precisely if \({{V}}\)  refines \({{U}}\) .
Let \({\mathrm {GCov}}(M)\)  denote the full subcategory of \({\mathrm {Cov}}(M)\)  on the differentiably good open coverings.
Note that \({\mathrm {Cov}}(M)\)  is cofiltered since every pair \(({{U}},{{V}})\)  of open coverings has a common refinement \({{U}}\times _M {{V}}\) .
Moreover, any open covering of \(M\)  has a differentiably good refinement [1]}, which implies that \({\mathrm {GCov}}(M)\)  is cofiltered and that the canonical inclusion \(j_M \colon {\mathrm {GCov}}(M) \hookrightarrow {\mathrm {Cov}}(M)\)  is final, since for any \({{U}}\in {\mathrm {Cov}}(M)\)  the slice category \(j/{{U}}\)  is cofiltered as well.
